Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data

Xinhua Zhao; Zilai Li

doi:10.3934/cpaa.2020052

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2020, Volume 19, Issue 3: 1421-1448. Doi: 10.3934/cpaa.2020052

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Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data

Xinhua Zhao^1, and
Zilai Li^2, ,

1.
School of Mathematics, South China University of Technology, Guangzhou 510641, China
2.
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China

^* Corresponding author
^* Corresponding author

Received: April 2019

Revised: June 2019

Published: March 2020

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Abstract

In this paper, we study the asymptotic behavior of global spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations for the viscous heat conducting ideal polytropic gas flow with large initial data in $ H^1 $, when the heat conductivity coefficient depends on the temperature, practically, $ \kappa(\theta) = \tilde{\kappa}_1+\tilde{\kappa}_2\theta^q $ where constants $ \tilde{\kappa}_1>0 $, $ \tilde{\kappa}_2>0 $ and $ q>0 $ (as to the case of $ \tilde{\kappa}_1 = 0 $, please refer to the Appendix). In addition, the exponential decay rate of solutions toward to the constant state as time tends to infinity for the initial boundary value problem in bounded domain is obtained. The mass density and temperature are proved to be pointwise bounded from below and above, independent of time although strong nonlinearity in heat diffusion. The analysis is based on some delicate uniform energy estimates independent of time.

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Mathematics Subject Classification: Primary: 35B40, 35Q30; Secondary: 76N10.

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